Automatically update an UIList
Thanks to @batFINGER and @poor comments, I've been able to do this, so I'm gonna post it here for if someone has the same problem.

The trick had to be done in the panel draw method, when drawing the UIList:

row.template_list("MyList", "", bpy.data, "actions", context.scene, "myListIndex")


The third and fourth arguments are the place from were the list items will be get. I was using a custom list here (that just pointed to an action), so that was redundant.
The fifth and sixth arguments are where the selected item index is stored, which in my case is in the Scene properties.

Thanks for your help!